Sequences and Series jee mains questions | Sequences and Series Previous Year Question

Q 31 :

Let $a_{1}, a_{2}, a_{3}, \dots$ be an arithmetic progression. If $\frac{a_{1} + a_{2} + \dots + a_{10}}{a_{1} + a_{2} + \dots + a_{P}} = \frac{100}{P^{2}}$ (where $P \neq 0$ ) then $\frac{a_{11}}{a_{10}} =$

$\frac{19}{21}$
$\frac{100}{21}$
$\frac{21}{19}$
$\frac{121}{100}$

1

2

3

4

(3)

$\frac{S_{10}}{S_{p}} = \frac{100}{p^{2}} \Rightarrow S_{p} = \frac{S_{10} p^{2}}{100}, \frac{a_{11}}{a_{10}} = \frac{S_{11} - S_{10}}{S_{10} - S_{9}} = \frac{21}{19}$

Q 32 :

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

87
88
89
90

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2

3

4

(3)

Use $a_{n}$ formula to find a = 4 and d = 5.

Q 33 :

If the ratio of the sum to $n$ terms of two A.P.'s is $(5 n + 3) : (3 n + 4)$ , then the ratio of their $4^{t h}$ terms is

$\frac{16}{23}$
$\frac{23}{16}$
$\frac{38}{25}$
$\frac{25}{38}$

1

2

3

4

(3)

Given $\frac{\frac{n}{2} [2 a_{1} + (n - 1) d_{1}]}{\frac{n}{2} [2 a_{2} + (n - 1) d_{2}]} = \frac{5 n + 3}{3 n + 4}$

For $n = 7, \frac{a_{1} + 3 d_{1}}{a_{2} + 3 d_{2}} = \frac{38}{25}$

Q 34 :

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is: [2026]

$\frac{5}{87}$
$\frac{25}{83}$
$\frac{15}{29}$
$\frac{5}{29}$

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2

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4

(1)

We have, $a = \frac{10}{3}$

Then, $l = \frac{10}{3} + 29 d$

Given, $S = l^{3}$

$\Rightarrow \frac{30}{2} [\frac{20}{3} + 29 d] = {(\frac{10}{3} + 29 d)}^{3}$

$\Rightarrow 15 [\frac{20}{3} + 29 d] = {(\frac{10}{3} + 29 d)}^{3}$

From options, $d = \frac{5}{87}$

Q 35 :

Let $A_{1}, A_{2}, A_{3}, . . ., A_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to: [2026]

129
136
131.5
134

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4

(4)

Given 39 arithmetic means between 59 and 159.

So, sequence is $59, A_{1}, A_{2}, A_{3}, . . ., A_{39}, 159$ , (4! terms)

Common difference, $d = \frac{159 - 59}{40} = \frac{100}{40} = \frac{5}{2}$

$∴ A_{k} = 59 + k \cdot (\frac{5}{2})$

$∴ A_{25} = 59 + 25 (\frac{5}{2}) = 121.5$

$A_{28} = 59 + 28 (\frac{5}{2}) = 129$

$A_{31} = 59 + 31 (\frac{5}{2}) = 136.5$

$A_{36} = 59 + 36 (\frac{5}{2}) = 149$

$∴ Mean = \frac{121.5 + 129 + 136.5 + 149}{4} = 134$

Q 36 :

Let $α$ = 3 + 4 + 8 + 9 + 13 + 14 + ... uto 40 terms. If ${(\tan β)}^{\frac{α}{1020}}$ is a root of the equation $x^{2} + x - 2 = 0$ , $β \in (0, \frac{π}{2})$ , then $\sin^{2} β + 3 \cos^{2} β$ is equal to: [2026]

2
$\frac{7}{4}$
$\frac{5}{2}$
$\frac{3}{2}$

1

2

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4

(1)

$α = 3 + 4 + 8 + 9 + 13 + 14 + . . . (40 terms)$

$α = (3 + 8 + 13 + . . . (20 terms) + (4 + 9 + 14 + . . . (20 terms)$

$\Rightarrow α = \frac{20}{2} [6 + 19 \times 5] + \frac{20}{2} [8 + 19 \times 5]$

$\Rightarrow α = 10 [101 + 103] = 2040$

Now, is a root of the equation.

$\Rightarrow x^{2} + x - 2 = 0$

$\Rightarrow x^{2} + 2 x - x - 2 = 0$

$\Rightarrow x = - 2, 1$

$∴ \tan^{2} β = 1 \Rightarrow β = \frac{π}{4}$

So, $\sin^{2} β + 3 \cos^{2} β = \frac{1}{2} + \frac{3}{2} = 2$

Q 37 :

The number of common terms in the progressions 4, 9, 14, 19, ...... up to 25th term and 3, 6, 9, 12, ...... up to 35th term is _____

9
5
7
8

1

2

3

4

(3)

in 1st, 25th term is = 124 in 2nd, 35th term is = 105

Common terms are 9, 24, 39, 54, 69, 84, 99

Common difference is L.C.M of 5, 3

Topic Question Set

Q 31 :

Let $a_{1}, a_{2}, a_{3}, \dots$ be an arithmetic progression. If $\frac{a_{1} + a_{2} + \dots + a_{10}}{a_{1} + a_{2} + \dots + a_{P}} = \frac{100}{P^{2}}$ (where $P \neq 0$ ) then $\frac{a_{11}}{a_{10}} =$

Q 32 :

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

Q 33 :

If the ratio of the sum to $n$ terms of two A.P.'s is $(5 n + 3) : (3 n + 4)$ , then the ratio of their $4^{t h}$ terms is

Q 34 :

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is: [2026]

Q 35 :

Let $A_{1}, A_{2}, A_{3}, . . ., A_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to: [2026]

Q 36 :

Let $α$ = 3 + 4 + 8 + 9 + 13 + 14 + ... uto 40 terms. If ${(\tan β)}^{\frac{α}{1020}}$ is a root of the equation $x^{2} + x - 2 = 0$ , $β \in (0, \frac{π}{2})$ , then $\sin^{2} β + 3 \cos^{2} β$ is equal to: [2026]

Q 37 :

The number of common terms in the progressions 4, 9, 14, 19, ...... up to 25th term and 3, 6, 9, 12, ...... up to 35th term is _____

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Topic Question Set

Q 31 : Let a1,a2,a3,… be an arithmetic progression. If a1+a2+⋯+a10a1+a2+⋯+aP=100P2 (where P≠0) then a11a10=

Q 32 : If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

Q 33 : If the ratio of the sum to n terms of two A.P.'s is (5n+3):(3n+4), then the ratio of their 4th terms is

Q 34 : The first term of an A.P. of 30 non-negative terms is 103. If the sum of this A.P. is the cube of its last term, then its common difference is: [2026]

Q 35 : Let A1, A2, A3, ..., A39 be 39 arithmetic means between the numbers 59 and 159. Then the mean of A25, A28, A31 and A36 is equal to: [2026]

Q 36 : Let α = 3 + 4 + 8 + 9 + 13 + 14 + ... uto 40 terms. If (tanβ)α1020 is a root of the equation x2+x–2=0, β∈(0,π2), then sin2β+3cos2β is equal to: [2026]

Q 37 : The number of common terms in the progressions 4, 9, 14, 19, ...... up to 25th term and 3, 6, 9, 12, ...... up to 35th term is _____

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Q 31 :

Let $a_{1}, a_{2}, a_{3}, \dots$ be an arithmetic progression. If $\frac{a_{1} + a_{2} + \dots + a_{10}}{a_{1} + a_{2} + \dots + a_{P}} = \frac{100}{P^{2}}$ (where $P \neq 0$ ) then $\frac{a_{11}}{a_{10}} =$

Q 32 :

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

Q 33 :

If the ratio of the sum to $n$ terms of two A.P.'s is $(5 n + 3) : (3 n + 4)$ , then the ratio of their $4^{t h}$ terms is

Q 34 :

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is: [2026]

Q 35 :

Let $A_{1}, A_{2}, A_{3}, . . ., A_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to: [2026]

Q 36 :

Let $α$ = 3 + 4 + 8 + 9 + 13 + 14 + ... uto 40 terms. If ${(\tan β)}^{\frac{α}{1020}}$ is a root of the equation $x^{2} + x - 2 = 0$ , $β \in (0, \frac{π}{2})$ , then $\sin^{2} β + 3 \cos^{2} β$ is equal to: [2026]

Q 37 :

The number of common terms in the progressions 4, 9, 14, 19, ...... up to 25th term and 3, 6, 9, 12, ...... up to 35th term is _____